Problem: $f(x, y) = x^3 + xy^2$ We have a change of variables: $\begin{aligned} x &= X_1(r, \theta) = r \cos(\theta) \\ \\ y &= X_2(r, \theta) = r \sin(\theta) \end{aligned}$ What is $f(x, y)$ under the change of variables? Choose 1 answer: Choose 1 answer: (Choice A) A $r^4\sin(\theta)$ (Choice B) B $r^3\sin(\theta)$ (Choice C) C $r^4\cos(\theta)$ (Choice D) D $r^3\cos(\theta)$
Answer: When applying a change of variables, we substitute the new definition for $x$ and $y$ into the original equation. The original equation: $f(x, y) = x^3 + xy^2$ Let's substitute $X_1(r, \theta)$ for $x$ and $X_2(r, \theta)$ for $y$. $\begin{aligned} f(x, y) &= r^3\cos^3(\theta) + r\cos(\theta)r^2\sin^2(\theta) \\ \\ &= r\cos(\theta) ( r^2\cos^2(\theta) + r^2\sin^2(\theta) ) \\ \\ &= r\cos(\theta) (r^2) \\ \\ &= r^3\cos(\theta) \end{aligned}$ Therefore, under the change of variables, $f(x, y)$ becomes: $r^3\cos(\theta)$